Why does my mesh enlarge upon applying automatic weights?
There are several issues here.

Unapplied scales

When rigging, every object's scale should be applied. if you select your rig and check its scale, you can see it isn't even nor equal to 1:


To correct that, select the object, hit CtrlA and select Apply Scale.

Wrong Parenting

Rigged models should be parented to your Rig object and the Armature modifiers should target that same Rig object. You have parented you model to a wrong object. And that wrong object also happens to have transforms on it, which influences your model, so we'll have to take care of that too.


Remove the armature modifier of your model.
With your model selected, hit AltP → Clear Parent and Keep Transforms.
Apply your model's scale.
Select your rigged model, then your Rig object, and hit CtrlP → Set parent to Armature Deform with Automatic Weights.


Now everything should be fine. You'll just have to work out those weights.